A modular form [1]} \(f : \mathbb {H}\rightarrow \mathbb {C}\)  of weight \(k\)  and level \(N\)  associated to a Dirichlet
characterA Dirichlet character (modulo \(N\) ) is a group homomorphism \(\chi _N\in \mathrm {Hom}(G(N),\mathbb {C}^*)\)  from the
multiplicative group \(G(N)=\left(\mathbb {Z}/N\mathbb {Z}\right)^*\)  of invertible elements of \(\mathbb {Z}/N\mathbb {Z}\)  to \(\mathbb {C}^*\) . It is a
multiplicative character. It customarily extended to all integers by setting \(\chi _N(m)=0\)  for all \(m\)  which share common factors with \(N\) 
[2]}. \(\chi _N\)  modulo \(N\) , is a holomorphic form on the upper half plane \(\mathbb {H}\)  that transforms, under the action of a
discrete subgroup \(\Gamma (N) \subset \Gamma \subset \Gamma (1)\) , as
\(f(\gamma z) \equiv f\left(\frac{az + b}{cz + d}\right) = \chi _N(d) (cz + d)^{k} f(z), \quad \) 
